distributed control for spatial self-organization of...
TRANSCRIPT
DISTRIBUTED CONTROL FOR SPATIAL SELF-ORGANIZATION1
OF MULTI-AGENT SWARMS∗2
VISHAAL KRISHNAN AND SONIA MARTINEZ †3
Key words. Self-organization, Distributed control, Pseudo-localization, Harmonic maps4
AMS subject classifications. 34B45, 35B40, 35B35, 58J32, 58J35, 58E205
Abstract. In this work, we design distributed control laws for spatial self-organization of multi-6agent swarms in 1D and 2D spatial domains. The objective is to achieve a desired density distribution7over a simply-connected spatial domain. Since individual agents in a swarm are not themselves of8interest and we are concerned only with the macroscopic objective, we view the network of agents in9the swarm as a discrete approximation of a continuous medium and design control laws to shape the10density distribution of the continuous medium. The key feature of this work is that the agents in11the swarm do not have access to position information. Each individual agent is capable of measuring12the current local density of agents and can communicate with its spatial neighbors. The network13of agents implement a Laplacian-based distributed algorithm, which we call pseudo-localization, to14localize themselves in a new coordinate frame, and a distributed control law to converge to the15desired spatial density distribution. We start by studying self-organization in one-dimension, which16is then followed by the two-dimensional case.17
1. Introduction. Self-organization in swarms refers broadly to the emergence18
of patterns of long-range order in large collectives of dynamic agents which interact19
locally with each other. Self-organization is a pervasive phenomenon in nature, ob-20
served in biological [6] and other natural systems [27]. This has greatly inspired the21
development of large scale robotic counterparts [23], with applications to monitoring,22
manipulation, and construction. This transition does not merely involve an increase in23
the size of robotic networks, but it also introduces new theoretical challenges for their24
analysis and control design. In particular, large groups of agents have some essen-25
tial characteristics that distinguish them from other smaller-scale counterparts. In a26
swarm, individual agents have no significance and only the macroscopic objectives are27
relevant. A swarm largely remains unaffected by the removal of a large, but discrete,28
number of agents. Moreover, it is difficult (and needlessly complicated) to specify29
the global configuration of the swarm using the states of individual agents; instead,30
employing macroscopic quantities such as the swarm spatial density distribution to31
specify its configuration is more appropriate. From an analysis and control-theoretic32
viewpoint, the dynamic modeling of swarms is less explored, which e.g. can be es-33
tablished by means of PDEs, for which control theoretic tools are less well developed34
in comparison to ODEs. These theoretical challenges motivate the investigation of35
self-organization in large-scale swarms.36
In the literature, Markov-chain based methods have been widely used in address-37
ing some of the key theoretical problems pertaining to swarm self-organization. By38
means of it, the swarm configuration is described through the partitioning the spatial39
domain in a finite number of larger size disjoint subregions, on which a probability40
distribution is defined. Then, the self-organization problem is reduced to the design41
of the transition matrix governing the evolution of this probability density function42
∗Submitted to the editors 20 June 2016. A preliminary version of this work is to appear in the22nd International Symposium on Mathematical Theory of Networks and Systems as [18].
Funding: This work has been partially supported by grant AFOSR-11RSL548.†The authors are with the Department of Mechanical and Aerospace Engineering, University of
California at San Diego, La Jolla CA 92093 USA (email: [email protected]; [email protected]).
1
This manuscript is for review purposes only.
to ensure its convergence to a desired profile. A recent approach to density control43
using Markov chains is presented in [10], which includes additional conflict-avoidance44
constraints. In this setting every agent is able to determine the bin to which it be-45
longs at every instant of time, which essentially means that individual agents have46
self-localization capabilities. Also, the dimensional transition matrix is synthesized47
in a central way at every instant of time by solving a convex optimization problem.48
In [3], the authors make use of inhomogeneous Markov chains to minimize the number49
of transitions to achieve a swarm formation. In this approach, the algorithm necessi-50
tates the estimation of the current swarm distribution, and computes the transition51
Markov matrices for each agent, at each instant of time. The fact that every agent52
needs to have an estimate of the global state (swarm distribution) at every time may53
not be desirable or feasible. The localization of each agent still remains to be a main54
assumption. Under similar conditions, one can find the manuscripts [1] and [7], which55
describe probabilistic swarm guidance algorithms. In [5], the authors present an ap-56
proach to task allocation for a homogeneous swarm of robots. This is a Markov-chain57
based approach, where the goal is to converge to the desired population distribution58
over the set of tasks.59
In the context of robotic swarms, programmable self-assembly of two-dimensional60
shapes with a thousand-robot swarm is demonstrated in [24]. These robots are capable61
of measuring distances to nearby neighbors which they use to localize themselves62
relative to other localized robots. Each robot then uses its position to implement an63
edge-following algorithm.64
Another approach uses partial differential equations to model swarm behaviour,65
and control action is applied along the boundary of the swarm. Previous works on66
PDE-based methods with boundary control include [14], where the authors present67
an algorithm for the deployment of agents onto families of planar curves. Here, the68
swarm collective dynamics are modeled by the reaction-advection-diffusion PDE and69
the particular family of curves to which the swarm is controlled to is parametrized by70
the continuous agent identity in the interval of unit length. An extension of this work71
to deployment on a family of 2D surfaces in 3D space can be found in [22]. More-72
over, in [13] the authors present a distributed optimal control problem formulation for73
swarm systems, where microscopic control laws are derived from the optimal macro-74
scopic description using a potential function approach. The problem of position-free75
extremum-seeking of an external scalar signal using a swarm of autonomous vehicles,76
inspired by bacterial chemotaxis, has been studied in [21].77
In this work, we adopt a viewpoint outlined in [2], wherein we make an amorphous78
medium abstraction of the swarm, which is essentially a manifold with an agent79
located at each point. We then model the system using PDEs and design distributed80
control laws for them. An important component of this paper is the Laplacian-81
based distributed algorithm which we call pseudo-localization algorithm, which the82
agents implement to localize themselves in a new coordinate frame. The convergence83
properties of the graph Laplacian to the manifold Laplacian have been studied in [4],84
which find useful applications in this paper.85
The main contribution of this paper is the development of distributed control laws86
for the index- and position-free density control of swarms to achieve general 1D and87
a large class of 2D density profiles. In very large swarms with thousands of agents,88
particularly those deployed indoors or at smaller scales, presupposing the availability89
of position information or pre-assignment of indices to individual agents would be a90
strong assumption. In this paper, in addition to not making the above assumptions,91
the agents are only capable of measuring the local density, and in the 2D case, the92
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This manuscript is for review purposes only.
density gradient and the normal direction to the boundary.93
Under these assumptions, we present distributed pseudo-localization algorithms94
for one and two dimensions that agents implement to compute their position identi-95
fiers. Since every agent occupies a unique spatial position, we are able to rigorously96
characterize the resulting position assignment as a one-to-one correspondence between97
the set of spatial coordinates and the set of position identifiers, which corresponds98
to a diffeomorphism of the continuum domain. Based on this assignment, we then99
design control strategies for self-organization in one and two dimensions under the100
assumption that the motion control of agents is noiseless. The extension to the 2D101
case leads to new difficulties related to the control of the swarm boundaries. To ad-102
dress these, we implement a variant of the 1D pseudo-localization algorithm at the103
boundary during an initialization phase. A preliminary version of this work appeared104
in [18] where we presented an outline of the algorithms and state some of the results.105
We develop them here rigorously, providing detailed proofs for our claims.106
The paper is organized as follows. In Section 2, we introduce the basic notation107
and preliminary concepts used in the manuscript. We present the analysis of self-108
organization in one dimension in Section 4, where we introduce the pseudo-localization109
algorithm in Section 4.1 and the distributed control law in Section 4.2. After this, we110
generalize and extend the analysis for self-organization in two dimensions in Section 5.111
Section 6 contains numerical simulations of the results in the paper, and in Section 7,112
we present our conclusions.113
2. Preliminaries. Let R denote the set of all real numbers, R≥0 the set of non-114
negative real numbers, and Rn the n-dimensional Euclidean space. We use boldface115
letters to denote vectors in Rn. The norm |x| of a vector x ∈ Rn is the standard116
Euclidean 2-norm, unless otherwise specified. Let ∇ =(
∂∂x1
, . . . ∂∂xn
)denote the117
gradient operator in Rn when acting on real-valued functions and the Jacobian in118
the context of vector-valued functions. As a shorthand, we let ∂∂z (·) = ∂z(·) for a119
variable z. Let ∆ =∑ni=1
∂2
∂x2i
be the Laplace operator in Rn. We denote by either120
S or dSdt the total time derivative of S(t). Given functions f, g : R → R, we write121
f = O(g) if there exist positive constants C and c such that |f(h)| ≤ C|g(h)|, for122
all |h| ≤ c. Let S denote the set of agents in the swarm, and N its cardinality. For123
the 1D case, let l ∈ S denote the leftmost agent, and r ∈ S the rightmost one. Let124
Ni denote the spatial neighborhood of agent i, which comprises those agents located125
inside a small ball centered at i. A set-valued mapping, denoted by f : R ⇒ R2,126
maps the set of real numbers onto subsets of R2. For a bounded open set Ω ⊂ Rn,127
∂Ω denotes its boundary, Ω = Ω ∪ ∂Ω its closure and Ω = Ω \ ∂Ω its interior with128
respect to the standard Euclidean topology. The set of smooth real-valued functions129
on Ω is denoted by C∞(Ω). We let µ (or dx in 1D) denote the standard Lebesgue130
measure; with a slight abuse of notation, we sometimes omit dµ (resp. dx in 1D) from131
long integrals. The Dirac measure δ on Ω defined for any x ∈ Ω and any measurable132
set D ⊆ Ω is given by δx(D) = 1 for x ∈ D, and δx(D) = 0 for x /∈ D.133
For two non-empty subsets M1 and M2 of a metric space (M,d), the Hausdorff134
distance dH(M1,M2) between them is defined as:135
dH(M1,M2) = max supx∈M1
infy∈M2
d(x, y), supy∈M2
infx∈M1
d(x, y).(1)136
137
The set of functions on a measurable space U , given by Lp(U) = f : U → R | ‖f‖Lp(U) =138(∫U|f |pdµ
)1/p< ∞, constitute the Lp space, where ‖ · ‖Lp(U) is the Lp norm. Of139
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This manuscript is for review purposes only.
particular interest is the L2 space, or the space of square-integrable functions. In140
this paper, we denote by ‖f‖L2(U) the L2 norm of f with respect to the Lebesgue141
measure, and by ‖f‖L2(U,ρ) the weighted L2 norm (with the strictly positive weight ρ142
on U). The Sobolev space W 1,p(U) over a measurable space U is defined as W 1,p(U) =143
f : U → R | ‖f‖W 1,p =(∫U|f |p +
∫U|∇f |p
)1/p< ∞. Of particular interest is the144
space W 1,2, also called the H1 space. For two functions f(t, ·) and g(·), we denote by145
f →L2 g the convergence in L2 norm (over the domain U of the functions) of f(t, ·)146
to g(·) as t → ∞, that is, limt→∞ ‖f(t, ·) − g(·)‖L2 = 0. Convergence in H1 norm is147
denoted similarly by f →H1 g.148
We now state some well-known results that we will be used in the subsequent149
sections of this paper.150
Lemma 2.1. (Divergence Theorem [9]). For a smooth vector field F over a151
bounded open set Ω ⊆ Rn with boundary ∂Ω, the volume integral of the divergence152
∇ · F of F over Ω is equal to the surface integral of F over ∂Ω:153
∫
Ω
(∇ · F) dµ =
∫
∂Ω
F · n dS,(2)154155
where n is the outward normal to the boundary and dS the measure on the boundary.156
For a scalar field U and a vector field F defined over Ω ⊆ Rn:157
∫
Ω
(F · ∇U) dµ =
∫
∂Ω
U(F · n) dS −∫
Ω
U(∇ · F) dµ.158159
Lemma 2.2. (Leibniz Integral Rule [9]). Let f ∈ C∞(R×Rn) and Ω : R ⇒ Rn160
be a smooth one-parameter family of bounded open sets in Rn generated by the flow161
corresponding to the smooth vector field v on Rn. Then:162
d
dt
(∫
Ω(t)
f(t, r) dµ
)=
∫
Ω(t)
∂t(f(t, r)) dµ+
∫
∂Ω(t)
f(t, r)v · n dS.163
164
Corollary 2.3. (Derivative of Energy Functional). Let U be an energy165
functional defined as follows:166
U =1
2
∫
Ω
|f |2 dµ,167168
for some function f : Ω→ R. Then,169
∂tU =
∫
Ω
f ·(df
dt
)dµ+
1
2
∫
Ω
|f |2∇ · v dµ.170171
where ddt = ∂t + v · ∇ is the total derivative.172
Proof. We have included the proof for this corollary for the sake of completeness.173
Using the Leibniz integral rule and the Divergence theorem, we have (it is understood174
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This manuscript is for review purposes only.
that the integrations are with respect to the measure µ):175
∂U
∂t=
∫
Ω
f · ft +1
2
∫
∂Ω
|f |2v · n176
=
∫
Ω
f · ft +1
2
∫
Ω
∇ · (|f |2v)177
=
∫
Ω
f · ft +
∫
Ω
f · (v · ∇)f +1
2
∫
Ω
|f |2∇ · v178
=
∫
Ω
f · (ft + (v · ∇)f) +1
2
∫
Ω
|f |2∇ · v179
=
∫
Ω
f ·(df
dt
)+
1
2
∫
Ω
|f |2∇ · v.180181
Lemma 2.4. (Poincare-Wirtinger Inequality [20]). For p ∈ [1,∞] and Ω, a182
bounded connected open subset of Rn with a Lipschitz boundary, there exists a constant183
C depending only on Ω and p such that for every function u in the Sobolev space184
W 1,p(Ω):185
‖u− uΩ‖Lp(Ω) ≤ C‖∇u‖Lp(Ω),186187
where uΩ = 1|Ω|∫
Ωudµ, and |Ω| is the Lebesgue measure of Ω.188
Lemma 2.5. (Rellich-Kondrachov Compactness Theorem [12]). Let U ⊂189
Rn be open, bounded and such that ∂U is C1. Suppose 1 ≤ p < n, then W 1,p(U) is190
compactly embedded in Lq(U) for each 1 ≤ q < pnn−p . Moreover, for [0, L] ⊂ R, the191
inclusion W 1,2([0, L]) ⊂ L2([0, L]) is also compact.192
Lemma 2.6. (LaSalle Invariance Principle [16, 26]). Let P(t) | t ∈ R≥0193
be a semigroup of nonlinear operators acting on U (closed subset of a Banach space194
with norm ‖ · ‖), and for any u ∈ U , define the positive orbit starting from u at t = 0195
as Γ+(u) = P(t)u | t ∈ R≥0 ⊆ U (we assume P(t) | t ∈ R≥0 to be such that the196
orbit Γ+(u) is smooth). Let V be a Lyapunov functional on U (such that V (u) ≤ 0197
in U). Define E = u ∈ U | V (u) = 0, and let E be the largest invariant subset198
of E. If for u0 ∈ U , the orbit Γ+(u0) is pre-compact (lies in a compact subset of U),199
then limt→+∞ d(P(t)u0, E) = 0, where d(y, E) = infx∈E ‖y − x‖.200
2.1. Continuum model of the swarm. Given that N , the number of agents201
in the swarm, is very large, we will analyze the swarm dynamics through a continuum202
approximation. Let t ∈ R≥0, and let M : R ⇒ Rn be a smooth one-parameter family203
of bounded open sets, such that the agents are deployed over M(t) at time t. We204
denote by ri(t) = vi, ∀i ∈ S, where ri(t) ∈ M(t) is the position of the ith agent in the205
swarm at time t. Let ρ : R≥0 × Rn → R≥0 be the spatial density function supported206
on M(t) for all t ≥ 0 (with ρ(t, r) > 0 for r ∈ M(t)), such that∫M(t)
ρ(t, r)dµ = 1.207
We assume that M(t) is simply connected and that the boundary ∂M(t) does not208
self-intersect for all t ≥ 0.209
Assuming that ρ is smooth, the macroscopic dynamics can now be described by210
the continuity equation [9], assuming that the total number of agents is conserved:211
∂ρ
∂t+∇ · (ρv) = 0, ∀ r ∈ M(t),(3)212
213
where v : R≥0 × Rn → Rn is the velocity field with vi(t) = v(t, ri), such that the214
one-parameter family M is generated by the flow associated with v.215
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This manuscript is for review purposes only.
2.2. Harmonic maps and diffeomorphisms. Let (M, g) and (N,h) be two216
Riemannian manifolds of dimensions m and n, and Riemannian metrics g and h,217
respectively. A map φ : M → N is called harmonic if it minimizes the functional:218
E(φ) =
∫
M
|dφ|2dvg,(4)219220
where dvg is the Riemannian volume form on M , and |dφ| is the Hilbert-Schmidt221
norm of dφ given at each point x ∈M , in local coordinates (x1, . . . , xm) on M , by:222
|dφx|2 = gij(x)hαβ(φ(x))∂φα
∂xi
∂φβ
∂xj.(5)223
224
Here, we use the Einstein summation convention, where a summation is implicit over225
repeated superscript-subscript pairs (i.e., kili ≡∑i kili). When g and h are both the226
Euclidean metric δ (where δij = 1 if i = j and 0 otherwise), we have:227
|dφx|2 =∑
α
∑
i
(∂φα
∂xi
)2
.(6)228
229
The Euler-Lagrange equation for the functional E, which also yields the minimum230
energy, is given by ∆φ = 0, the Laplace equation [17]. It is useful to note that the231
solutions to the heat equation, in the limit t→∞, approach the harmonic map. This232
is proved later in Lemma 5.1, and forms the basis for the design of the distributed233
pseudo-localization algorithm. We now state a lemma on harmonic diffeomorphisms234
of Riemann surfaces (i.e., m = n = 2 above).235
Lemma 2.7. (Harmonic diffeomorphism [11]). Let (M, g) be a compact sur-236
face with boundary and (N,h) a compact surface with non-positive curvature. Suppose237
that ψ : M → N is a diffeomorphism onto ψ(M). Assume that ψ(M) is convex.238
Then there is a unique harmonic map φ : M → N with φ = ψ on ∂M , such that239
φ : M → φ(M) is a diffeomorphism.240
We note that the non-positive curvature constraint in the lemma is essentially a241
constraint on the metric h on N , and the curvature is zero for the Euclidean metric.242
3. Problem description and conceptual approach. In this section, we pro-243
vide a high-level description of the proposed problem and explain the conceptual idea244
behind our approach. The technical details can be found in the following sections.245
The problem at hand is to ultimately design a distributed control law for a swarm246
to converge to a desired configuration. Here, a swarm configuration is a density247
function ρ of the multi-agent system and the objective is that agents reconfigure248
themselves into a desired known density ρ∗. To do this, an agent at position x is able249
to measure the current local density value, ρ(t, x); however, its position x within the250
swarm is unknown. Thus, given ρ∗, an agent at x cannot directly compute ρ∗(x) nor251
a feedback law based on ρ − ρ∗. To solve this problem, we devise a mechanism that252
allows agents to determine their coordinates in a distributed way in an equivalent253
coordinate system.254
Note that, given a diffeomorphism Θ∗ from the spatial domain of the swarm onto255
the unit interval or disk (i.e. a coordinate transformation), we can equivalently pro-256
vide the agents with a transformed density function p∗, such that p∗ = ρ∗ (Θ∗)−1.257
In this way, instead of ρ∗ the agents are given p∗, but still do not have access to Θ∗.258
The pseudo-localization algorithm is a mechanism that agents employ to progressively259
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This manuscript is for review purposes only.
compute an appropriate (configuration-dependent) diffeomorphism by local interac-260
tions.261
In 1D, the pseudo-localization algorithm is a continuous-time PDE system in262
a new variable or pseudo-coordinate X which plays the role of an “approximate x263
coordinate” that agents can use to know where they are. The input to this system is264
the current density value ρ, see Figure 1 for an illustration, and the objective is that265
X converges to a ρ-dependent diffeomorphism. On the other hand, the variable X266
and the function p∗ are used to define the control input of another PDE system in the267
density ρ. In this way, we have a feedback interconnection of two systems, one in X268
and one in ρ, with the goal to achieve X → Θ∗ (the pseudo-coordinate X converges269
to a true coordinate given by Θ∗) and ρ→ ρ∗.
!tX = G(X, ")
!t" = F (", X, p!)
!X ! !" (coordinates)
! ! !" (objective)
Fig. 1: Feedback interconnection of pseudo-localization system in X and system in ρin the 1D case. The function p∗ is an equivalent density objective provided to agentsin terms of a diffeomorphism Θ∗. The variables X play the role of coordinates andeventually converge to the true coordinates given by Θ∗. Agents use p∗ and X tocompute the control in the equation ρ. In turn, agents move and this will require are-computation of coordinates or update in X. The control strategy in the 2D case(stages 2 and 3) can be interpreted similarly.
270As for the control design methodology, we broadly follow a constructive, Lyapunov-271
based approach to designing distributed control laws for the swarm dynamics modeled272
by PDEs. For this, we define appropriate non-negative energy functionals that en-273
code the objective and choose control laws that keep the time derivative of the energy274
functional non-positive. This, along with well-known results on the precompactness275
of solutions as in Lemma 2.5, the Rellich Kondrachov compactness theorem, allows us276
to apply the LaSalle Invariance Principle in Lemma 2.6 and other technical arguments277
to establish the convergence results that we seek.278
In the 1D case, we can identify a set of diffeomorphisms Θ associated with any279
ρ that eventually converge to Θ∗, and simultaneously control boundary agents into280
a desired final domain (the support of ρ∗). These are given by the cumulative dis-281
tribution function associated with the density function; see Section 4.1. The 2D case282
is more complex, and analogous results could not be derived in their full generality.283
First, unlike the 1D case, a cumulative distribution does not lead to a diffeomorphism284
in general. Instead, we set out to find diffeomorphisms as the result of a distributed285
algorithm. Given that the discretization of heat flow naturally leads to distributed286
algorithms, we investigate under what conditions this is the case via harmonic map287
theory. On the control side, there also are additional difficulties, and because of this,288
we simplify the control strategy into three stages. In the first stage, the boundary289
agents are re-positioned onto the boundary of the desired domain while containing290
the others in the interior. Once this is achieved, the second and third stages can be291
seen again as the interconnection of two systems in pseudo-coordinates R = (X,Y )292
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This manuscript is for review purposes only.
(instead of X) and ρ, analogously to Figure 1. However, we apply a two time-scale293
separation for analysis by which coordinates are computed in a fast-time scale and294
reconfiguration is done in a slow-time scale, which allows for a sequential analysis of295
the two stages. We then study the robustness of this approach.296
4. Self-organization in one dimension. In this section, we present our pro-297
posed pseudo-localization algorithm and the distributed control law for the 1D self-298
organization problem.299
Mathematically, for each t ∈ R≥0, letM(t) = [0, L(t)] ⊂ R be the interval in which300
the agents are distributed in 1D, and let ρ : R× R→ R≥0 be the normalized density301
function supported on M(t), for all t ≥ 0 (with ρ(t, x) > 0, ∀x ∈M(t)), describing the302
swarm on that interval. Without loss of generality, we place the origin at the leftmost303
agent of the swarm. We also assume that the leftmost and the rightmost agents, l304
and r, are aware that they are at the boundary. Let ρ∗ : M∗ = [0, L∗]→ R>0 be the305
desired normalized density distribution.306
Since a direct feedback control law can not be implemented by agents because307
they do not have access to their positions, we introduce an equivalent representation308
of the density ρ∗, p∗, depending on a particular diffeomorphism Θ∗. First, define309
Θ∗ : M∗ → [0, 1] such that Θ∗(x) =∫ x
0ρ∗(x)dx and Θ∗(L∗) = 1.310
Now, let p∗ : [0, 1] → R>0, and θ∗ ∈ Θ∗(M∗) = [0, 1], be such that p∗(θ∗) =311
ρ∗((Θ∗)−1(θ∗)) = ρ∗(x).312
ρ∗(x) = p∗(θ∗)
x ∈ [0, L∗] Θ∗(x) = θ∗ ∈ [0, 1]
ρ∗
Θ∗
p∗
The function p∗, which represents the desired density distribution mapped onto313
the unit interval [0, 1], is computed offline and is broadcasted to the agents prior to314
the beginning of the self-organization process. We use p∗ to derive the distributed315
control law which the agents implement. We assume that p∗ is a Lipschitz function316
in the sequel.317
4.1. Pseudo-localization algorithm in one dimension. We first consider318
the static case, that is, the design of the pseudo-localization dynamics on X of the319
upper block in Figure 1, when the agents and ρ are stationary. We define Θ : M =320
[0, L]→ [0, 1] as:321
Θ(x) =
∫ x
0
ρ(x)dx,(7)322323
such that Θ(L) = 1. In other words, Θ is the cumulative distribution function (CDF)324
associated with ρ. (Note that the domains are static and hence the argument t has325
been dropped, which will be reintroduced later.)326
Lemma 4.1. (The CDF diffeomorphism). Given ρ : M → R>0 a smooth327
function, the mapping Θ : M → [0, 1] as defined above, is a diffeomorphism and328
Θ(M) = [0, 1].329
Proof. Since ρ(x) > 0, ∀x ∈M , it follows that Θ is a strictly increasing function330
of x, and is therefore a one-to-one correspondence on M . Moreover, Θ is smooth331
and has a differentiable inverse, which implies it is a diffeomorphism. Finally, since332
Θ(L) = 1, we have Θ(M) = [0, 1].333
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This manuscript is for review purposes only.
Our goal here is to set up a partial differential equation with appropriate boundary334
conditions that yield the diffeomorphism Θ as its asymptotically stable steady-state335
solution. We begin by setting up the pseudo-localization dynamics for a stationary336
swarm (for which the spatial domain M and the density distribution ρ are fixed). Let337
X : R×M → R be such that (t, x) 7→ X(t, x) ∈ R, with:338
∂tX =1
ρ∂x
(∂xX
ρ
),
X(t, 0) = α(t),
X(t, L) = β(t),
∂tα(t) = −α(t),
∂tβ(t) = 1− β(t),
X(0, x) = X0(x),
(8)339
340
where α : R→ R is a control input at the boundary x = 0 and β : R→ R is a control341
input at the boundary x = L. From (7), we observe that ∂x
(∂xΘρ
)= 0. Letting342
w = X −Θ denote the error, we obtain:343
∂tw =1
ρ∂x
(∂xw
ρ
),
w(t, 0) = α(t),
w(t, L) = β(t)− 1,
∂tw(t, 0) = −w(t, 0),
∂tw(t, L) = −w(t, L),
w(0, x) = w0(x) = X0(x)−Θ(x).
(9)344
345
346
Assumption 4.2. (Well-posedness of the pseudo-localization dynamics).347
We assume that the pseudo-localization dynamics (8) (and (9)) is well-posed, that348
the solution is sufficiently smooth (at least C2 in the spatial variable, even as t→∞)349
and belongs to the Sobolev space H1(M) for every t ∈ R≥0.350
Lemma 4.3. (Pointwise convergence to diffeomorphism). Under Assump-351
tion 4.2, on the well-posedness of the pseudo-localization dynamics, and for bounded ρ,352
the solutions to PDE (8) converge pointwise to the CDF diffeomorphism Θ defined in353
(7), as t→∞, for all smooth initial conditions X0.354
Proof. We prove that the solutions to the PDE (8) converge pointwise to the355
diffeomorphism Θ by showing that w → 0, as t→∞, pointwise for (9). For this, we356
consider a functional V , given by (integrations are taken with respect to the Lebesgue357
measure):358
V =1
2
∫
M
ρ|w|2 +1
2
∫
M
1
ρ|∂xw|2.359
360
The time derivative V is given by:361
V =
∫
M
ρw(∂tw) +
∫
M
1
ρ(∂xw)(∂t∂xw).362
363
9
This manuscript is for review purposes only.
Here, replace ∂tw in the first integral with the dynamics in (9), and then use ∂t∂x =364
∂x∂t in the second integral together with the Divergence Theorem in Lemma 2.1. We365
obtain:366
V =
∫
M
w∂x
(∂xw
ρ
)−∫
M
∂x
(∂xw
ρ
)∂tw +
∂xw
ρ∂tw
∣∣∣∣L
− ∂xw
ρ∂tw
∣∣∣∣0
367
= −∫
M
1
ρ|∂xw|2 −
∫
M
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
+w + ∂tw
ρ∂xw
∣∣∣∣L
− w + ∂tw
ρ∂xw
∣∣∣∣0
.368369
(After the second equal sign, apply again the Divergence Theorem on the first integral370
of the previous line, and replace ∂tw from (9).) Substituting from (9), we have:371
V = −∫
M
1
ρ|∂xw|2 −
∫
M
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
.372373
Clearly, V ≤ 0, and w(t, ·) ∈ H1(M), for all t. By the Rellich-Kondrachov Com-374
pactness Theorem of Lemma 2.5, H1(M) is compactly contained in L2(M). Thus,375
by the LaSalle Invariance Principle of Lemma 2.6, the solution to (9) converges to376
the largest invariant subset of V −1(0). Note that V = 0 implies∫M
1ρ |∂xw|2 = 0.377
Thus, we have limt→∞∫M
1ρ |∂xw|2 = 0. Since ρ is bounded (sup ρ < ∞), we have378
limt→∞1
sup ρ
∫M|∂xw|2 ≤ limt→∞
∫M
1ρ |∂xw|2 = 0, which implies limt→∞
∫M|∂xw|2 =379
limt→∞ ‖∂xw‖2L2(M) = 0. Now, limt→∞ |w(t, x)| = limt→∞ |w(t, 0) +∫ x
0∂xw(t, ·)| ≤380
limt→∞ |w(t, 0)|+∫ x
0|∂xw(t, ·)| ≤ limt→∞ |w(t, 0)|+
√L(t)‖∂xw(t, ·)‖L2(M) = 0 (since381
limt→∞ w(t, 0) = 0 and limt→∞ ‖∂xw(t, ·)‖L2(M) = 0). Thus, limt→∞ w(t, x) = 0, for382
all x ∈ M . Therefore, the solutions to (9) converge to w ≡ 0 pointwise, as t → ∞,383
from any smooth initial w0 = X0 −Θ.384
We now have that the solution to the pseudo-localization dynamics converges to385
the diffeomorphism Θ in the stationary case. For the dynamic case, we modify (8) to386
account for agent motion. Let X : R × R → R be supported on M(t) = [0, L(t)] for387
all t ≥ 0. Using the relation dXdt = ∂tX + v∂xX, where v is the velocity field on the388
spatial domain, we consider:389
∂tX =1
ρ∂x
(∂xX
ρ
)− v∂xX,
X(t, 0) = 0,
X(t, L(t)) = β(t),
X(0, x) = X0(x).
(10)390
391
In the dynamic case, and w.l.o.g. we have set α(t) = 0 for all t ≥ 0, for simplicity. We392
will use the above PDE system in the design of the distributed motion control law,393
redesigning the boundary control β to achieve convergence of the entire system. We394
now discretize (10) to obtain a distributed pseudo-localization algorithm. Let Xi(t) =395
X(t, xi), where xi ∈ M(t) is the position of the ith agent. We identify the agent i396
with its desired coordinate in the unit interval at time t, i.e., Θ(t, x) = θ ∈ [0, 1],397
where Θ(t, x) =∫ x
0ρ(t, x)dx from (7), which now shows the time dependency of ρ.398
In this way, ρ(t, x) = ∂xΘ(t, x). It follows that ∂x(·) = ∂θ(·)∂xθ = ∂θ(·)ρ. Therefore,3991ρ∂x(·) = ∂θ(·). From (10), we have:400
dX
dt= ∂tX + v∂xX =
1
ρ∂x
(∂xX
ρ
)= ∂θ (∂θX) =
∂2X
∂θ2.(11)401
402
10
This manuscript is for review purposes only.
Now, we discretize (11) with the consistent finite differences dXdt ≈
Xi(t+1)−Xi(t)∆t and403
∂2X∂θ2 ≈
Xi+1−2Xi+Xi−1
(∆θ)2 (that is, we have that lim∆t→0Xi(t+1)−Xi(t)
∆t = dXdt and that404
lim∆θ→0Xi+1−2Xi+Xi−1
(∆θ)2 = ∂2X∂θ2 ). Now, with the choice 3∆t = (∆θ)2, and from (10),405
we obtain for i ∈ S \ l, r:406
Xi(t+ 1) =1
3(Xi−1(t) +Xi(t) +Xi+1(t)) ,
Xl(t) = 0,
Xr(t) = β(t),
Xi(0) = X0i.
(12)407
408
Equation (12) is the discrete pseudo-localization algorithm to be implemented syn-409
chronously by the agents in the swarm, starting from any initial condition X0. The410
leftmost agent holds its value at zero while the rightmost agent implements the bound-411
ary control β. In the following section we analyze its behavior together with that of412
the dynamics on ρ.413
4.2. Distributed density control law and analysis. In this subsection, we414
propose a distributed feedback control law to achieve ρ→ ρ∗ and w → 0, as t→∞,415
through a distributed control input v and a boundary control β. We refer the reader to416
[19] for an overview of Lyapunov-based methods for stability analysis of PDE systems.417
From (3) and (10), we have the dynamics:418
∂tρ = −∂x(ρv),
∂tX =1
ρ∂x
(∂xX
ρ
)− v∂xX,
X(t, 0) = 0,
X(t, L(t)) = β(t),
X(0, x) = X0(x).
(13)419
420
This realizes the feedback interconnection of Figure 1.421
Assumption 4.4. (Well-posedness of the full PDE system). We assume422
that (13) is well posed, and that the solution ρ(t, ·) (resp. X(t, ·)) is sufficiently smooth423
and belongs to the Sobolev space H1([0, L(t)]), for all t ∈ R≥0 (resp. X belongs to424
the Sobolev space H1(M(t)) for all t ∈ R≥0).425
We also assume that the agent at position x at time t is able to measure ρ(t, x).426
However, the agents in the swarm do not have access to their positions, and therefore427
cannot access ρ∗(x), which could be used to construct a feedback law. To circumvent428
this problem, we propose a scheme in which the agents use the position identifier or429
pseudo-localization variable X to compute p∗ X(t, x), using this as their dynamic430
set-point. The idea is to then design a distributed control law and a boundary control431
law such that ρ→ p∗ X and X → Θ∗, as t→∞, to obtain ρ→ p∗ Θ∗ = ρ∗. Recall432
that the function p∗ is computed offline and is broadcasted to the agents prior to the433
beginning of the self-organization process, and that p∗ is assumed to be a Lipschitz434
function. Consider the distributed control law, defined as follows for all time t:435
v(t, 0) = 0,
∂xv = (ρ− p∗ X)− ∂Xp∗
ρ(ρ+ p∗ X)∂x
(∂xX
ρ
),
(14)436
437
11
This manuscript is for review purposes only.
together with the boundary control law:438
X(t, 0) = 0,
βt = k
(2− β(t)− Xx
ρ
∣∣∣∣L(t)
).
(15)439
440
We remark again that the agents implementing the control laws (14) and (15) do not441
require position information, because for the agent at position x at time t, ρ(t, x) is a442
measurement, X(t, x) is the pseudo-localization variable, through which p∗ X(t, x)443
can be computed.444
Theorem 4.5. (Convergence of solutions). Under the well-posedness As-445
sumption 4.4, the solutions (ρ(t, ·), X(t, ·)) to (13), under the control laws (14) and446
(15), converge to (ρ∗,Θ∗), ρ → ρ∗ and X → Θ∗ pointwise, as t → ∞, from any447
smooth initial condition (ρ0, X0).448
Proof. Consider the candidate control Lyapunov functional V :449
V =1
2
∫ L(t)
0
|ρ− p∗ X|2dx+1
2
∫ L(t)
0
|∂xw|2ρ
dx+1
2|w(L(t))|2.450
451
Taking the time derivative of V along the dynamics (13), using Lemma 2.2 on the452
Leibniz integral rule, and applying Corollary 2.3 on the derivative of energy function-453
als, we obtain:454
V =
∫ L(t)
0
(ρ− p∗ X)
(dρ
dt− d(p∗ X)
dt
)dx+
1
2
∫ L(t)
0
|ρ− p∗ X|2∂xv dx455
+
∫ L(t)
0
(∂xw)(∂t∂xw)
ρdx− 1
2
∫ L(t)
0
(∂xw
ρ
)2
(∂tρ)dx+1
2
(∂xw)2
ρv
∣∣∣∣L(t)
0
456
+ w(L)dw(L(t))
dt.457
458
Now, dρdt = ∂tρ+ v∂xρ = −ρ∂xv (since ∂tρ = −∂x(ρv), from (13)). Also, ∂t∂x = ∂x∂t,459
which implies that∫ L(t)
0(∂xw)(∂t∂xw)
ρ dx =∫ L(t)
0(∂xw)(∂x∂tw)
ρ dx = (∂xw)(∂tw)ρ
∣∣∣∣L(t)
0
−460
−∫ L(t)
0∂x
(∂xwρ
)(∂tw)dx (using the Divergence theorem in the second integral), and461
we obtain:462
V =
∫ L(t)
0
(ρ− p∗ X)
[−ρ∂xv − ∂Xp∗
1
ρ∂x
(∂xX
ρ
)]dx463
+1
2
∫ L(t)
0
|ρ− p∗ X|2∂xv dx+∂xw
ρ∂tw
∣∣∣∣L(t)
0
−∫ L(t)
0
∂x
(∂xw
ρ
)(∂tw)dx464
+1
2
∫ L(t)
0
(∂xw
ρ
)2
∂x(ρv)dx+1
2
(∂xw)2
ρv
∣∣∣∣L(t)
0
+ w(L)dw(L(t))
dt.465
466
From (13), we have that ∂tw = 1ρ∂x
(∂xwρ
)− v∂xw, thus:467
∫ L(t)
0
∂x
(∂xw
ρ
)(∂tw)dx =
∫ L(t)
0
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
dx−∫ L(t)
0
∂x
(∂xw
ρ
)(∂xw)vdx.468
469
12
This manuscript is for review purposes only.
Now, using the above equation, applying the Divergence theorem (2) (integration by470
parts) to the term 12
∫ L(t)
0
(∂xwρ
)2
∂x(ρv)dx, and rearranging the terms, we obtain:471
V =− 1
2
∫ L(t)
0
(ρ− p∗ X)
[(ρ+ p∗ X)(∂xv) +
∂Xp∗
ρ∂x
(∂xX
ρ
)]dx472
−∫ L(t)
0
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
dx+
∫ L(t)
0
∂x
(∂xw
ρ
)(∂xw)vdx473
−∫ L(t)
0
(∂xw)∂x
(∂xw
ρ
)vdx+
∂xw
ρ∂tw
∣∣∣∣L(t)
0
+(∂xw)2
ρv
∣∣∣∣L(t)
0
+ w(L)dw(L(t))
dt.474
475
Since ∂xwρ ∂tw
∣∣∣∣L(t)
0
+ (∂xw)2
ρ v
∣∣∣∣L(t)
0
= ∂xwρ (∂tw + v∂xw)
∣∣∣∣L(t)
0
= ∂xwρ
dwdt
∣∣∣∣L(t)
0
, the above476
equation reduces to:477
V =− 1
2
∫ L(t)
0
(ρ− p∗ X)
[(ρ+ p∗ X)(∂xv) +
∂Xp∗
ρ∂x
(∂xX
ρ
)]dx478
−∫ L(t)
0
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
dx+
(∂xw
ρ+ w
)dw
dt
∣∣∣∣L(t)
0
.479
480
From (14) and (15), we have dwdt
∣∣∣∣0
= 0 and dwdt
∣∣∣∣L(t)
= −k(∂xwρ + w
) ∣∣∣∣L(t)
, and we481
obtain:482
V =− 1
2
∫ L(t)
0
(ρ− p∗ X)
[(ρ+ p∗ X)(∂xv) +
∂Xp∗
ρ∂x
(∂xX
ρ
)]dx483
−∫ L(t)
0
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
dx− k∣∣∣∣∂xw
ρ+ w
∣∣∣∣2
L(t)
.484
485
With ∂xv = (ρ− p∗ X)− ∂Xp∗
ρ(ρ+p∗X)∂x
(∂xXρ
)as in (14), we get:486
V =− 1
2
∫ L(t)
0
(ρ+ p∗ X)|ρ− p∗ X|2dx−∫ L(t)
0
1
ρ
∣∣∣∣∂x(∂xw
ρ
)∣∣∣∣2
dx
− k∣∣∣∣∂xw
ρ+ w
∣∣∣∣2
L(t)
.
(16)487
488
Clearly, V ≤ 0, and ρ(t, ·), w(t, .) ∈ H1([0, supt L(t)]), for all t. By Lemma 2.5, the489
Rellich-Kondrachov Compactness Theorem, the space H1([0, supt L(t)]) is compactly490
contained in L2([0, supt L(t)]), and by the LaSalle Invariance Principle, Lemma 2.6,491
we have that the solutions to (13) converge to the largest invariant subset of V −1(0).492
This implies that:493
limt→∞
‖ρ(t, ·)− p∗ X(t, ·)‖L2([0,L(t)]) = 0,494
limt→∞
‖∂x(∂xw
ρ
)‖L2([0,L(t)],ρ) = 0,495
limt→∞
(∂xw
ρ
∣∣∣∣L(t)
+ w(t, L(t))
)= 0.496
497
13
This manuscript is for review purposes only.
Also, w(t, 0) = 0 and, from the smoothness of w, we have w(t, x) =∫ x
0∂xw. From498
above, we have limt→∞ ‖∂x(∂xwρ
)‖L2([0,L(t)],ρ) = 0, and using the Poincare-Wirtinger499
inequality, Lemma 2.4 (with the weighted measure ρdµ), we get limt→∞ ‖∂xwρ −500∫ L(t)
0∂xw‖L2([0,L(t)],ρ) = 0. Now
∫ L(t)
0∂xw = w(t, L(t)) and from above we have501
limt→∞ w(t, L(t)) = limt→∞−∂xwρ∣∣∣∣L(t)
, which implies that:502
limt→∞
∥∥∥∥∂xw
ρ+∂xw
ρ(t, L(t))
∥∥∥∥L2([0,L(t)],ρ)
= 0.503
504
It can be shown from above that limt→∞
∥∥∥∂xwρ∥∥∥L2([0,L(t)],ρ)
= limt→∞
∣∣∣∂xwρ (t, L(t))∣∣∣,505
and that the Cauchy-Schwarz inequality for the (weighted) inner product of the func-506
tions ∂xwρ (t, ·) and ∂xw
ρ (t, L(t)) in the limit t→∞ is indeed an equality. This implies:507
limt→∞
∣∣∣∣∂xw
ρ(t, ·)
∣∣∣∣ = limt→∞
∣∣∣∣∂xw
ρ(t, L(t))
∣∣∣∣508509
almost everywhere in [0, L(t)]. Owing to the smoothness of w, we therefore have510
limt→∞∂xwρ (t, ·) = limt→∞
∂xwρ (t, L(t)) a.e., and we get:511
limt→∞
∥∥∥∥∂xw
ρ
∥∥∥∥L2([0,L(t)],ρ)
= limt→∞
‖∂xw‖L2([0,L(t)]) = 0.512
513
Using the Poincare-Wirtinger inequality, Lemma 2.4, again, we note that this implies514
limt→∞ ‖w −∫ L(t)
0w‖L2([0,L(t)]) = 0. We have limt→∞ |
∫ L(t)
0w| = |
∫ L(t)
0
∫ x0∂xw| ≤515
L(t)3/2‖∂xw‖L2([0,L(t)]) = 0, which implies that limt→∞∫ L(t)
0w = 0 and therefore516
limt→∞ ‖w‖L2([0,L(t)]) = 0. Thus, we get limt→∞ ‖w(t, ·)‖H1([0,L(t)]) = 0, or in517
other words, w →H1 0. Now, limt→∞ |w(t, x)| = limt→∞ |w(t, 0) +∫ x
0∂xw(t, ·)| ≤518
limt→∞ |w(t, 0)|+∫ x
0|∂xw(t, ·)| ≤ limt→∞ |w(t, 0)|+
√L(t)‖w(t, ·)‖H1(M) = 0, which519
implies that w → 0 pointwise. Given that w = X − Θ, we have limt→∞X(t, ·) −520
Θ(t, ·) = 0. Let limt→∞ L(t) = L and limt→∞Θ(t, ·) = Θ(·), which implies that521
X → Θ pointwise.522
Now, from the above we have limt→∞ ‖ρ(t, ·)−p∗Θ‖L2([0,L(t)]) = limt→∞ ‖ρ(t, ·)−523
p∗ X(t, ·)+p∗ X(t, ·)−p∗ Θ‖L2([0,L(t)]) ≤ limt→∞ ‖ρ(t, ·)−p∗ X(t, ·)‖L2([0,L(t)]) +524
‖p∗ X(t, ·) − p∗ Θ‖L2([0,L(t)]) = 0 (this follows from the assumption that p∗ is525
Lipschitz, since ‖p∗ X − p∗ Θ‖L2 ≤ c‖X − Θ‖L2 for some Lipschitz constant c).526
Thus, we have ρ→L2 p∗ Θ.527
Now, we are interested in the limit density distribution ρ = p∗ Θ, and by the528
definition of Θ we have Θ(x) =∫ x
0ρ. We now prove that this limit (ρ, Θ) is unique, and529
that (ρ, Θ) = (ρ∗,Θ∗). From the definition of Θ, we get dΘdx (x) = ρ(x) = p∗(Θ(x)) > 0,530
∀Θ(x) ∈ [0, 1]. We therefore have:531
x =
∫ Θ(x)
0
(p∗(θ))−1dθ.532
533
Recall from the definition of p∗ and (7) that p∗ Θ∗(x) = ρ∗(x), and ddxΘ∗(x) =534
ρ∗(x) = p∗ Θ∗(x), which implies that dΘ∗
dx = p∗(θ∗) > 0, where θ∗ = Θ∗(x). There-535
14
This manuscript is for review purposes only.
fore:536
x =
∫ Θ∗(x)
0
(p∗(θ))−1dθ.537
538
From the above two equations, we get:539
∫ Θ(x)
0
(p∗(θ))−1dθ =
∫ Θ∗(x)
0
(p∗(θ))−1dθ,540
541
for all x, and since p∗ is strictly positive, it implies that Θ = Θ∗, and we obtain542
ρ = p∗ Θ = p∗ Θ∗ = ρ∗. And we know that ρ →L2 p∗ Θ = p∗ Θ∗ = ρ∗. In543
other words, ρ converges to ρ∗ in the L2 norm. Moreover, since X → Θ∗ pointwise,544
from (14) we have limt→∞ ∂xv = limt→∞ ρ − p∗ X = limt→∞ ρ − ρ∗, therefore545
limt→∞ ‖∂xv‖L2([0,L(t)]) = 0. Now, from the smoothness of v, we have:546
limt→∞
|v(t, x)| ≤ limt→∞
|v(t, 0)|+∫ x
0
|∂xv| ≤ limt→∞
|v(t, 0)|+√L(t)‖∂xv‖L2([0,L(t)]) = 0.547
548
Thus, limt→∞ ρ(t, x) − ρ∗(x) = limt→∞ v(t, x) = 0 pointwise, that is, ρ → ρ∗ point-549
wise. Therefore, for the PDE system (13), with control laws (14) and (15), we have550
ρ→ ρ∗ and X → Θ∗ (pointwise).551
4.2.1. Physical interpretation of the density control law. For a physical552
interpretation of the control law, we first rewrite some of the terms in a suitable form.553
From (13), we know that:554
1
ρ∂x
(∂xX
ρ
)=∂X
∂t+ v∂xX =
dX
dt.555
556
The second term in the expression for ∂xv in the law (14) can thus be rewritten as:557
∂Xp∗
ρ(ρ+ p∗ X)∂x
(∂xX
ρ
)=
1
(ρ+ p∗ X)∂Xp
∗ dX
dt=
1
(ρ+ p∗ X)
dp∗
dt.558
559
Now, from above and (14), we obtain:560
v(t, x) =
∫ x
0
(ρ− p∗ X)−∫ x
0
1
(ρ+ p∗ X)
dp∗
dt.(17)561
562
Equation (17) gives the velocity of the agent at x at time t. Now, to interpret it,563
we first consider the case where the pseudo-localization error is zero, that is, when564
X = Θ∗. This would imply that p∗ X = p∗ Θ∗ = ρ∗, dXdt = dΘ∗
dt = 0, and we obtain:565
v(t, x) =
∫ x
0
(ρ− ρ∗).(18)566567
The term∫ x
0(ρ− ρ∗) =
∫ x0ρ−
∫ x0ρ∗ is the difference between the number of agents in568
the interval [0, x] and the desired number of agents in [0, x]. If the term is positive, it569
implies that there are more than the desired number of agents in [0, x] and the control570
law essentially exerts a pressure on the agent to move right thereby trying to reduce571
the concentration of agents in the interval [0, x], and, vice versa, when the term is572
negative. This eventually accomplishes the desired distribution of agents over a given573
15
This manuscript is for review purposes only.
interval. This would be the physical interpretation of the control law for the case574
where the pseudo-localization error is zero (that is, the agents have full information575
of their positions).576
However, in the transient case when the agents do not possess full information577
of their positions and are implementing the pseudo-localization algorithm for that578
purpose, the control law requires a correction term that accounts for the fact that the579
transient pseudo coordinates X(t, x) cannot be completely relied upon. This is what580
the second term∫ x
01
(ρ+p∗X)dp∗
dt in (17) corrects for. When this term is positive, that581
is,∫ x
01
(ρ+p∗X)dp∗
dt > 0, it roughly implies that the “estimate” of the desired number582
of agents in the interval [0, x] is increasing (indicating that an increase in the concen-583
tration of agents in [0, x] is desirable), and the term essentially reduces the “rightward584
pressure” on the agent (note that this term will have a negative contribution to the585
velocity (17)).586
4.3. Discrete implementation. In this section, we present a scheme to com-587
pute p∗ (the transformed desired density profile) and a consistent discretization scheme588
for the distributed control law. We follow that up with a discussion on the convergence589
of the discretized system and a pseudo-code for the implementation.590
4.3.1. On the computation of p∗. In this subsection, we provide a means of591
computing p∗ from a given ρ∗ via interpolation. Let the desired domain M∗ = [0, L∗]592
be discretized uniformly to obtain M∗d = 0 = x1, . . . , xm = L∗ such that xj−xj−1 =593
h (constant step-size). Note that m is the number of interpolation points, not equal594
to the number of agents. The desired density ρ∗ : [0, L∗] → R>0 is known, and we595
compute the value of ρ∗ on M∗d to get ρ∗(x1, . . . , xm) = (ρ∗1, . . . , ρ∗m). We also have596
Θ∗(x) =∫ x
0ρ∗dµ, for all x ∈ [0, L∗]. Now, computing the integral with respect to the597
Dirac measure for the set M∗d , we obtain Θ∗d(x1, . . . , xm) = (θ∗1 , . . . , θ∗m), where θ∗1 = 0598
and θ∗k = 12
∑kj=1(ρ∗j−1+ρ∗j )h, for k = 2, . . . ,m (note that 0 = θ∗1 ≤ θ∗2 ≤ . . . ≤ θ∗m ≤ 1599
and limh→0 θ∗m = Θ∗(L∗) = 1). Now, the value of the function p∗ at any X ∈ [0, 1] can600
be now obtained from the relation p∗(θ∗k) = ρ∗k, for k = 1, . . . ,m, by an appropriate601
interpolation.602
(ρ∗1, . . . , ρ∗m) = p∗(θ∗1 , . . . , θ
∗m)
(x1, . . . , xm) (θ∗1 , . . . , θ∗m)
ρ∗
Θ∗
p∗
4.3.2. Discrete control law. A discretized pseudo-localization algorithm is603
given by (12). We now discretize (14) to obtain an implementable control law for a604
finite number of agents i ∈ S, and a numerical simulation of this law is later presented605
in Section 6.606
Let i ∈ S \ l, r. First note that ∂xv = (∂θv)
∣∣∣∣θ=Θ(x)
(∂xΘ) = (∂θv)
∣∣∣∣θ=Θ(x)
ρ607
(where v ≡ v(Θ(x))). Using a consistent backward differencing approximation, and608
recalling that ∆θ = ε, we can write:609
(∂xv)i ≈ ρivi − vi−1
∆θ= ρi
vi − vi−1
ε, i ∈ S610
611
where ρi is agent i’s density measurement.612
16
This manuscript is for review purposes only.
From Section 4.1, recall the consistent finite-difference approximation:613
1
ρ∂x
(∂xX
ρ
)
i
≈ 1
ε2(Xi−1 − 2Xi +Xi+1).614
615
With κ = 12ε , from (14) and the above equation, we obtain the law for agent i as:616
vi = vi−1 +ρi − p∗(Xi)
2κρi− 2κ
ρi(ρi + p∗(Xi))
(p∗(Xi+1)− p∗(Xi−1)
Xi+1 −Xi−1
)
× (Xi−1 − 2Xi +Xi+1)
(19)617
618
with vl = 0. The computation in v can be implemented by propagating from the619
leftmost agent to the rightmost agent along a line graph Gline (with message receipt620
acknowledgment). Note that this propagation can alternatively be formulated by621
each agent averaging appropriate variables with left and right neighbors, which will622
result in a process similar to a finite-time consensus algorithm. Now, the boundary623
control (15) is discretized (with ∂tβ ≈ β(t+1)−β(t)∆t ), with the choice k = 1
ε to:624
β(t+ 1) = β(t) + k∆t(2− β(t)− 2κ (β(t)−Xr−1(t)))
=4− 2ε
3β(t) +
1
3Xr−1(t)
(20)625
626
4.3.3. On the convergence of the discrete system. The discretized pseudo-627
localization algorithm (12) with the boundary control law (15), can be rewritten as:628
X(t+ 1) = X(t)− 1
3LX(t) + u(t),(21)629
630
where X(t) = (Xl(t), . . . , Xr(t)), L is the Laplacian of the line graph Gline and the631
input u(t) =(0, . . . , 0, ε3 (2− β(t))
). This discretized system is stable and we thereby632
have that the discretized pseudo-localization algorithm is consistent and stable. Thus,633
by the Lax Equivalence Theorem [25], the solution of (21) converges to the solution634
of (10) with the boundary control (15) as N → ∞. Due to the nonlinear nature of635
the discrete implementation of the equation in ρ, we are only certain that we have a636
consistent discrete implementation in this case (no similar convergence theorem exists637
for discrete approximations of nonlinear PDEs.)638
17
This manuscript is for review purposes only.
Algorithm 1 Self-organization algorithm for 1D environments
1: Input: ρ∗, K (number of iterations), ∆t (time step)2: Requires:3: Offline computation of p∗ as outlined in Section 4.3.14: Initialization Xi(0) = X0i, vi = 05: Leftmost and rightmost agents, l, r, resp., are aware they are at boundary6: for k := 1 to K do7: if i = l then8: agent l holds onto Xl(k) = 0 and vl(k) = 09: else if agent i ∈ l + 1, . . . , r − 1 then
10: agent i receives Xi−1(k) and Xi+1(k) from its left and right neighbors11: agent i implements the update (12)12: else if i = r then13: agent r receives Xr−1(k) from its left neighbor14: agent r implements the update (20)
15: for i := l to r do16: agent i computes velocity vi from (19)
17: agent i moves to xi(k + 1) = xi(k) + vi(k)∆t
5. Self-organization in two dimensions. In this section, we present the two-639
dimensional self-organization problem. Although our approach to the 2D problem is640
fundamentally similar to the 1D case, we encounter a problem in the two-dimensional641
case that did not require consideration in one dimension, and it is the need to control642
the shape of the spatial domain in which the agents are distributed. We overcome643
this problem by controlling the shape of the domain with the agents on the boundary,644
while controlling the density distribution of the agents in the interior.645
Let M : R ⇒ R2 be a smooth one-parameter family of bounded open subsets646
of R2, such that M(t) is the spatial domain in which the agents are distributed at647
time t ≥ 0. Let ρ : R× R2 → R≥0 be the spatial density function with support M(t)648
for all t ≥ 0; that is, ρ(t, x) > 0, ∀x ∈ M(t), and t ≥ 0. Without loss of generality,649
we shift the origin to a point on the boundary of the family of domains, such that650
(0, 0) ∈ ∂M(t), for all t. Let ρ∗ : M∗ → R>0 be the desired density distribution,651
where M∗ is the target spatial domain. From here on, we view M as a one-parameter652
family of compact 2-submanifolds with boundary of R2. Just as in the 1D case, the653
agents do no have access to their positions but know the true x- and y-directions.654
In what follows we present our strategy to solve this problem, which we divide655
into three stages for simplicity of presentation and analysis. In the first stage, the656
agents converge to the target spatial domain M∗ with the boundary agents controlling657
the shape of the domain. In stage two, the agents implement the pseudo-localization658
algorithm to compute the coordinate transformation. In the third stage, the boundary659
agents remain stationary and the agents in the interior converge to the desired density660
distribution. This simplification is performed under the assumption that, once the661
agents have localized themselves at a given time, they can accurately update this in-662
formation by integrating their (noiseless) velocity inputs. Noisy measurements would663
require that these phases are rerun with some frequency; e.g. using fast and slow time664
scales as described in Section 3.665
5.1. Pseudo-localization algorithm for boundary agents. To begin with,666
we propose a pseudo-localization algorithm for the boundary agents which allows for667
18
This manuscript is for review purposes only.
their control in the first stage. To do this, we assume that the agents have a boundary668
detection capability (can approximate the normal to the boundary), the ability to669
communicate with neighbors immediately on either side along the boundary curve,670
and can measure the density of boundary agents.671
Let M0 ⊂ R2 be a compact 2-manifold with boundary ∂M0 and let (0, 0) ∈ ∂M0.672
To localize themselves, the agents on ∂M0 implement the distributed 1D pseudo-673
localization algorithm presented in Section 4.1. This yields a parametrization of the674
boundary Γ : ∂M0 → [0, 1), with Γ(0, 0) = 0, such that the closed curve which is675
the boundary ∂M0 is identified with the interval [0, 1). We have that, for γ ∈ [0, 1),676
Γ−1(γ) ∈ ∂M0. For γ ∈ [0, 1), let s(γ) be the arc length of the curve ∂M0 from677
the origin, such that s(0) = 0 and limγ→1 s(γ) = l. We assume that the boundary678
agents have access to the unit outward normal n(γ) to the boundary, and thus the679
unit tangent s(γ).680
Let q : [0, l)→ R>0 denote the normalized density of agents on the boundary, such681
that we have∫ l
0q(s)ds = 1. Now the 1D pseudo-localization algorithm of Section 4.1682
serves to provide a 2D boundary pseudo-localization as follows. Note that dsdγ = 1
q(γ) ,683
and (dx, dy) = sds, which implies (dx, dy) = 1q(γ)s(γ)dγ. Therefore, we get the684
position of the boundary agent at γ, (x(γ), y(γ)), as (x(γ), y(γ)) =∫ γ
01
q(γ)s(γ)dγ,685
and the arc-length s(γ) =∫ γ
01
q(γ)dγ, which is discretized by a consistent scheme to686
obtain:687
(xi, yi) =1
2∆γ
i−1∑
k=0
(skqk
+sk+1
qk+1
), for i ∈ ∂M0,(22)688
689
and we recall that the agents have access to q and s. The computation of (xi, yi)690
can be implemented by propagating from the agent with γi = 0 along the boundary691
agents in the direction as γi → 1, along a line graph Gline (with message receipt692
acknowledgment). Note that this propagation can alternatively be formulated by693
each agent averaging appropriate variables with left and right neighbors, which will694
result in a process similar to a finite-time consensus algorithm.695
This way, the boundary agents are localized at time t = 0, and they update their696
position estimates using their velocities, for t ≥ 0.697
5.2. Pseudo-localization algorithm in two dimensions. In this subsection,698
we present the pseudo-localization algorithm for the agents in the interior of the spatial699
domain. We first describe the idea of the coordinate transformation (diffeomorphism)700
we employ and construct a PDE that converges asymptotically to this diffeomorphism.701
We then discretize the PDE to obtain the distributed pseudo-localization algorithm.702
The main idea is to employ harmonic maps to construct a coordinate trans-703
formation or diffeomorphism from the spatial domain of the swarm onto the unit704
disk. We begin the construction with the static case, where the agents are station-705
ary. Let M ⊆ R2 be a compact, static 2-manifold with boundary and N = (x, y) ∈706
R2 | (x− 1)2 + y2 ≤ 1 be the unit disk. The manifolds M and N are both equipped707
with a Euclidean metric g = h = δ.708
First, we define a mapping for the boundary of M . Let Γ : ∂M → [0, 1) be a709
parametrization of the boundary of M , as outlined in Section 5.1. Let ξ : M → N be710
any diffeomorphism that takes the following form on the boundary of M :711
ξ(Γ−1(γ)) = (1− cos(2πγ), sin(2πγ)), γ ∈ [0, 1),(23)712713
19
This manuscript is for review purposes only.
and we know that Γ−1[0, 1) = ∂M .714
Now, from Lemma 2.7, on harmonic diffeomorphisms, there is a unique harmonic715
diffeomorphism, Ψ : M → N , such that Ψ = ξ on ∂M . We know that, by definition,716
the mapping Ψ = (ψ1, ψ2) satisfies:717
∆ψ1 = 0,
∆ψ2 = 0,for r ∈ M,
Ψ = ξ, on ∂M,
(24)718
719
where ∆ is the Laplace operator. Let Ψ∗ be the corresponding map from the target720
domain M∗ to the unit disk N . Now, we define a function p∗ : N → R>0 by p∗ =721
ρ∗ (Ψ∗)−1, the image of the desired spatial density distribution on the unit disk,722
which is computed offline and is broadcasted to the agents prior to the beginning of723
the self-organization process. We later use p∗ to derive the distributed control law724
which the agents implement.725
ρ∗(r) = p∗(Ψ∗(r))
r ∈M∗ Ψ∗(r) ∈ N
ρ∗
Ψ∗
p∗
We now construct a PDE that asymptotically converges to the harmonic diffeo-726
morphism, which we then discretize to obtain a distributed pseudo-localization algo-727
rithm. We use the heat flow equation as the basis to define the pseudo-localization728
algorithm, which yields a harmonic map as its asymptotically stable steady-state so-729
lution. We begin by setting up the system for a stationary swarm, for which the730
spatial domain is fixed.731
Let M ⊂ R2 be a compact 2-manifold with boundary, N be the unit disk of R2,732
and R = (X,Y ) : M → N . The heat flow equation is given by:733
∂tX = ∆X,
∂tY = ∆Y,for r ∈ M,
R = ξ, on ∂M.
(25)734
735
The heat flow equation has been studied extensively in the literature. For well-known736
existence and uniqueness results, we refer the reader to [11].737
Lemma 5.1. (Pointwise convergence of the heat flow equation to a har-738
monic diffeomorphism). The solutions of the heat flow equation (25) converge739
pointwise to the harmonic map satisfying (24), exponentially as t → ∞, from any740
smooth initial R0 ∈ H1(M)×H1(M).741
Proof. Let Ψ be the solution to (24), which is a harmonic map by definition. Let742
R = R−Ψ be the error where R = (X,Y ) is the solution to (25). Subtracting (24)743
from (25), we obtain:744
∂tX = ∆X,
∂tY = ∆Y,for r ∈ M,
R = 0, on ∂M.
(26)745
746
20
This manuscript is for review purposes only.
The Laplace operator ∆ with the Dirichlet boundary condition in (26) is self-adjoint747
and has an infinite sequence of eigenvalues 0 < λ1 < λ2 < . . ., with the corresponding748
eigenfunctions φi∞i=1 forming an orthonormal basis of L2(M) (where φi ∈ L2(M)749
and ∆φi = λiφi for all i, with φi = 0 on the boundary) [12]. Let the initial con-750
dition be X0 =∑∞i=1 aiφi and Y0 =
∑∞i=1 biφi (where ai and bi are constants751
for all i). The solution to (26) is then given by X(t, r) =∑∞i=1 aie
−λitφi(r) and752
Y (t, r) =∑∞i=1 bie
−λitφi(r). Since λi > 0, for all i, we obtain limt→∞ X(t, r) = 0 and753
limt→∞ Y (t, r) = 0, for all r ∈ M . Therefore, limt→∞R(t, r) = Ψ(r), for all r ∈ M ,754
and the convergence is exponential.755
We now have a PDE that converges to the diffeomorphism given by (24) for the756
stationary case (agents in the swarm are at rest). For the dynamic case, and to757
describe the algorithm while the agents are in motion, we modify (25) as follows. Let758
R = (X,Y ) : R×R2 → R. We are only interested in the restriction to M(t), R|M(t),759
at any time t, so we drop the restriction and just identify R ≡ R|M(t). Using the760
relation dXdt = ∂tX +∇X · v, where v is a velocity field, we obtain:761
∂tX = ∆X −∇X · v,∂tY = ∆Y −∇Y · v, for r ∈ M(t),
R = ξ, on ∂M(t).
(27)762
763
We now discretize (27) to derive the distributed pseudo-localization algorithm. Now,764
we have ρ : R× R2 → R≥0 with support M(t), the density distribution of the swarm765
on the domain M(t). We view the swarm as a discrete approximation of the domain766
M(t) with density ρ, and the PDE (27) as approximated by a distributed algorithm767
implemented by the swarm.768
Here, we propose a candidate distributed algorithm, which would yield the heat769
flow equation via a functional approximation. Our candidate algorithm is a time-770
varying weighted Laplacian-based distributed algorithm, owing to the connection be-771
tween the graph Laplacian and the manifold Laplacian [4]:772
Xi(t+ 1) = Xi(t) +∑
j∈Ni(t)
wij(t)(Xj(t)−Xi(t)),(28)773
774
and a similar equation for Y . We show how to derive next the values for the weights775
wij(t) ∈ R, for all t. First, the set of neighbors, j ∈ Ni(t), of i at time t, are the spatial776
neighbors of i in M(t), that is, Ni(t) = j ∈ S | ‖rj(t)−ri(t)‖ ≤ ε ≡ Bε(ri(t)). Using777
Xi(t+ 1)−Xi(t) = dXdt δt, for a small δt, we make use of a functional approximation778
of (28):779
dX
dtδt =
∫
Bε(ri(t))
w(t, ri, s)(X(t, s)−X(t, ri)) ρ(t, s)dµ,(29)780
781
where dν = ρ dµ is a density-dependent measure on the manifold, and the weighting782
function w satisfies w(t, ri(t), rj(t)) = wij(t), for all i, j ∈ S. We note that the783
summation term in (28) is a special form of the integral in (29) with a Dirac measure784
dν supported on the set r1(t), . . . , rN (t) at time t. Now, with the choice w(t, ri, s) =7851∫
Bε(s(t))ρ(t,s)dµ
and for very small ε (making O(ε3) terms negligible), (29) reduces to:786
dX
dtδt = a∆X,787
788
21
This manuscript is for review purposes only.
where a = 14ε
∫Bε(ri(t))
(s − ri(t)) · (s − ri(t))dµ is a constant. Now, with the choice789
δt = a, we obtain:790
dX
dt=∂X
∂t+ v · ∇X = ∆X,791
792
which is the PDE (27). Let d(t, ri(t)) =∫Bε(ri(t))
ρ(t, s)dµ and di(t) = |Ni(t)|, for793
i ∈ S. Substituting wij(t) = w(t, ri(t), rj(t)) = 1∫Bε(rj(t))
ρ(t,s)dµ= 1
d(t,rj(t))≈ 1
dj(t),794
in (28), we get the distributed pseudo-localization algorithm for the agents in the795
interior of the swarm to be:796
Xi(t+ 1) = Xi(t) +∑
j∈Ni(t)
1
dj(t)(Xj(t)−Xi(t)),
Yi(t+ 1) = Yi(t) +∑
j∈Ni(t)
1
dj(t)(Yj(t)− Yi(t)).
(30)797
798
For the agents on the boundary ∂M(t), we have:799
Ri = (Xi, Yi) = ξi,800801
where ξi = ξ(ri(t)), for ri(t) ∈ ∂M(t). Note that the discretization scheme is consis-802
tent, in that as the number of agents N →∞, the discrete equation (30) converges to803
the PDE (27). In this way, from (30), the pseudo-localization algorithm is a Laplacian-804
based distributed algorithm, with a time-varying weighted graph Laplacian.805
5.3. Distributed density control law and analysis. In this section, we de-806
rive the distributed feedback control law to converge to the desired density distribution807
over the target domain in the two-dimensional case. The swarm dynamics are given808
by:809
∂tρ = −∇ · (ρv), for r ∈ M(t),
∂tr = v, on ∂M(t).(31)810
811
812
Assumption 5.2. (Well-posedness of the PDE system). We assume that (31)813
is well-posed, and that its solution ρ(t, ·) is sufficiently smooth and belongs to the814
Sobolev space H1(M(t)), for all t ∈ R≥0.815
In what follows, we describe the control strategy based on three different stages.816
5.3.1. Stage 1. In this stage, the objective is for the swarm to converge to the817
target spatial domain M∗.818
Let r∗ : [0, 1] → ∂M∗ be the closed curve describing the desired boundary. Let819
e(γ) = r(γ) − r∗(γ) be the position error of agent γ on the boundary, where r(γ)820
is the actual position of agent γ computed as presented in Section 5.1. We define a821
distributed control law for swarm motion as follows:822
v = −∇ρρ , for r ∈ M(t),
∂tv = −e− v, on ∂M(t).(32)823
824
825
22
This manuscript is for review purposes only.
Theorem 5.3. (Convergence to the desired spatial domain). Under the826
well-posedness Assumption 5.2, the domain M(t) of the system (31), with the dis-827
tributed control law (32) converges to the target spatial domain M∗ as t → ∞, from828
any initial domain M0 with smooth boundary.829
Proof. We consider an energy functional E given by:830
E =1
2
∫
∂M(t)
|e|2 +1
2
∫
∂M(t)
|v|2.831
832
Its time derivative, E, using (32), is given by:833
E =
∫
∂M(t)
e · v +
∫
∂M(t)
v · ∂tv =
∫
∂M(t)
(e + v) · ∂tv = −∫
∂M(t)
|v|2.834
835
Clearly, E ≤ 0, and |v(t, ·)| ∈ H1(∪tM(t)), for all t. By Lemma 2.5, the Rellich-836
Kondrachov Compactness theorem, H1(∪tM(t)) is compactly contained in the space837
L2(∪tM(t)) and by the LaSalle Invariance Principle, Lemma 2.6, we have that the838
solutions to (31) with the control law (32) converge to the largest invariant subset839
of E−1(0), which satisfies:840
limt→∞
‖|v|‖L2(∂M(t)) = 0,841
limt→∞
∂t‖|v|‖L2(∂M(t)) = limt→∞
∫
∂M(t)
v · ∂tv = 0.842
843
The set E−1(0) is characterized by the first equality above and the second equality844
is further satisfied by the invariant subset of E−1(0). We know from (32) that ∂tv =845
−e−v on ∂M(t), which upon multiplying on both sides by v, integrating over ∂M(t)846
and applying the previous equality on the integral of v·∂tv, yields limt→∞∫∂M(t)
e·v =847
0. Now, we have |∂tv|2 = |e|2 + |v|2 + 2e · v, which on integrating over ∂M(t) yields848
limt→∞ ‖|∂tv|‖L2(∂M(t)) = limt→∞ ‖|e|‖L2(∂M(t)). By multiplying ∂tv = −e − v on849
both sides by ∂tv, integrating over ∂M(t), and using the Cauchy-Schwarz inequality,850
we obtain:851
limt→∞
‖|∂tv|‖2L2(∂M(t)) = limt→∞
−∫
∂M(t)
e · ∂tv ≤ limt→∞
∫
∂M(t)
|e||∂tv|852
≤ limt→∞
‖|e|‖L2(∂M(t))‖|∂tv|‖L2(∂M(t)) = limt→∞
‖|∂tv|‖2L2(∂M(t))853854
In this way, the Cauchy-Schwarz inequality becomes an equality, which implies that855
limt→∞∫∂M(t)
[|e||∂tv| − (−e) · ∂tv] = 0 (since the integrand is non-negative and its856
integral is zero, it is zero almost everywhere), thus limt→∞ ∂tv = − limt→∞ e almost857
everywhere (a.e.) on the boundary, and, in turn, implies that limt→∞ v = 0 a.e. on858
the boundary (since ∂tv = −e − v and limt→∞ ∂tv = − limt→∞ e). From here, and859
owing to the Invariance Principle, we have limt→∞ ∂tv = 0 = limt→∞ e a.e. on the860
boundary. Thus, we have that limt→∞M(t) = M∗.861
5.3.2. Stage 2. Here, the agents in the swarm implement the pseudo-localization862
algorithm presented in Section 5.2. Since the agents are distributed across the target863
spatial domain M∗, implementing the pseudo-localization algorithm yields the coordi-864
nate transformation Ψ∗ characteristic of the domain M∗. We therefore have ∂tΨ∗ = 0,865
which implies that dΨ∗
dt = ∂tΨ∗ +∇(Ψ∗)v = ∇(Ψ∗)v, which will be used in Stage 3.866
23
This manuscript is for review purposes only.
5.3.3. Stage 3. In this stage, the boundary agents of the swarm remain station-867
ary and interior agents converge to the desired density distribution.868
Consider the distributed control law, defined as follows for all time t:869dvdt = −ρ∇(ρ− p∗ Ψ∗) + (v · ∇)v − v, for r ∈ M∗,v = 0, on ∂M∗,
(33)870
871
where dvdt at r ∈M is the acceleration of the agent at r, the control input. Using the872
relation ddt = ∂t + v · ∇, it follows from (33) that ∂tv = −ρ∇(ρ− p∗ Ψ∗)− v.873
Theorem 5.4. (Convergence to the desired density). The solutions ρ(t, ·)874
to (31) for the fixed domain M∗, under the distributed control law (33) and the well-875
posedness Assumption 5.2, converge to the desired density distribution ρ∗ a.e. as t→876
∞, from any smooth initial condition ρ0.877
Proof. We consider an energy functional E given by:878
E =1
2
∫
M∗|ρ− p∗ Ψ∗|2 +
1
2
∫
M∗|v|2.879
880
Using Corollary 2.3, to compute the derivative of energy functionals, we obtain E881
(letting ∇ = (∂X , ∂Y )) as follows:882
E =
∫
M∗(ρ− p∗ Ψ∗)
(dρ
dt− d(p∗ Ψ∗)
dt
)+
1
2
∫
M∗|ρ− p∗ Ψ∗|2∇ · v
+
∫
M∗v · ∂tv
= −∫
M∗(ρ− p∗ Ψ∗)
(ρ∇ · v + ∇p∗ · dΨ∗
dt
)+
1
2
∫
M∗|ρ− p∗ Ψ∗|2∇ · v
+
∫
M∗v · ∂tv
= −1
2
∫
M∗(ρ2 − (p∗ Ψ∗)2)∇ · v −
∫
M∗(ρ− p∗ Ψ∗)∇p∗ · dΨ∗
dt+
∫
M∗v · ∂tv,
883
884
where, to obtain the third equality, we expand the square |ρ− p∗ Ψ∗|2 in the second885
integral of the second equality. Since v = 0 on ∂M∗ and from Section 5.3.2, we have886dΨ∗
dt = ∇(Ψ∗)v, we obtain:887
E =1
2
∫
M∗∇(ρ2 − (p∗ Ψ∗)2) · v −
∫
M∗(ρ− p∗ Ψ∗)∇p∗ · (∇Ψ∗v) +
∫
M∗v · ∂tv.888
889
We have ∇p∗∇Ψ∗ = ∇(p∗ Ψ∗), and ∇(ρ2 − (p∗ Ψ∗)2) = (ρ − p∗ Ψ∗)∇(ρ + p∗ 890
Ψ∗) + (ρ+ p∗ Ψ∗)∇(ρ− p∗ Ψ∗). Thus, we get:891
E =1
2
∫
M∗(ρ+ p∗ Ψ∗)∇(ρ− p∗ Ψ∗) · v +
1
2
∫
M∗(ρ− p∗ Ψ∗)∇(ρ+ p∗ Ψ∗) · v
−∫
M∗(ρ− p∗ Ψ∗)∇(p∗ Ψ∗) · v +
∫
M∗v · ∂tv.
892
893
We now have:894
E =1
2
∫
M∗(ρ+ p∗ Ψ∗)∇(ρ− p∗ Ψ∗) · v
+1
2
∫
M∗(ρ− p∗ Ψ∗)∇(ρ− p∗ Ψ∗) · v +
∫
M∗v · ∂tv.
895
896
24
This manuscript is for review purposes only.
We therefore get:897
E =
∫
M∗ρ∇(ρ− p∗ Ψ∗) · v +
∫
M∗v · ∂tv =
∫
M∗v · (ρ∇(ρ− p∗ Ψ∗) + ∂tv) .898
899
From (33), we have ∂tv = −ρ∇(ρ− p∗ Ψ∗)− v, and we obtain:900
E = −∫
M∗|v|2.901
902
Clearly, E ≤ 0, and ρ(t, .) ∈ H1(M∗) for all t. By Lemma 2.5, the Rellich-Kondrachov903
Compactness theorem, H1(M∗) is compactly contained in L2(M∗), and by the Invari-904
ance Principle, Lemma 2.6, we have that the solution to (31) converges to the largest905
invariant subset of E−1(0), which satisfies:906
‖|v|‖L2(M∗) = 0,
1
2∂t‖|v|‖2L2(M∗) =
∫
M∗v · ∂tv = 0.
(34)907
908
The set E−1(0) is characterized by the first equality above and the second equality is909
further satisfied by the invariant subset of E−1(0). We know from (33) that910
∂tv = −ρ∇(ρ− p∗ Ψ∗)− v,(35)911912
which substituted in (34) yields∫M∗ ρv · ∇(ρ − p∗ Ψ∗) = 0. Now, from (35), we913
obtain ‖|∂tv|‖2L2(M∗) =∫M∗ |ρ∇(ρ−p∗ Ψ∗)|2 +
∫M∗ |v|2 +2
∫M∗ ρv ·∇(ρ−p∗ Ψ∗) =914 ∫
M∗ |ρ∇(ρ − p∗ Ψ∗)|2; that is, ‖|∂tv|‖L2(M∗) = ‖|ρ∇(ρ − p∗ Ψ∗)|‖L2(M∗). By915
multiplying (35) by ∂tv on both sides and applying the Cauchy-Schwarz inequality,916
we can also get that ‖|∂tv|‖2L2(M∗) = −∫M∗ ρ∂tv · ∇(ρ− p∗ Ψ∗) ≤
∫M∗ |∂tv||ρ∇(ρ−917
p∗Ψ∗)| ≤ ‖|∂tv|‖L2(M∗)||ρ∇(ρ−p∗Ψ∗)|‖L2(M∗) = ‖|∂tv|‖2L2(M∗). Thus, the Cauchy-918
Schwarz inequality is in fact an equality, which implies that ∂tv = −ρ∇(ρ− p∗ Ψ∗)919
almost everywhere in M∗, which, from (35) implies in turn that v = 0 a.e. in M∗. It920
thus follows that ∂tv = 0 and ∇(ρ− p∗ Ψ∗) = 0 a.e in M∗, and therefore ρ− p∗ Ψ∗921
is constant a.e. in M∗. Using the Poincare-Wirtinger inequality, Lemma 2.4, we922
obtain that ‖(ρ − p∗ Ψ∗) − (ρ − p∗ Ψ∗)M∗‖ ≤ C‖∇(ρ − p∗ Ψ∗)‖ = 0, where923
(ρ− p∗ Ψ∗)M∗ = 1|M∗|
∫M∗(ρ− p∗ Ψ∗). Since
∫M∗ ρ =
∫Np∗ =
∫M∗ p
∗ Ψ∗ = 1, we924
have that (ρ−p∗ Ψ∗)M∗ = 0, and therefore ‖ρ−p∗ Ψ∗‖L2(M∗) = 0. Now, combined925
with the fact that ρ − p∗ Ψ∗ is constant a.e. in M∗, we obtain that ρ = p∗ Ψ∗926
a.e. in M∗. We know that p∗ Ψ∗ = ρ∗ and therefore, ρ = p∗ Ψ∗ = ρ∗, which is the927
desired density distribution. Thus, limt→∞ ρ = ρ∗ a.e. in M∗.928
5.3.4. Robustness of the distributed control law. The self-organization929
algorithm in 2D has been divided into three stages, where asymptotic convergence is930
achieved in each stage (with exponential convergence in the second stage). We now931
present a robustness result for convergence in Stage 3 under incomplete convergence932
in the preceding stages.933
Lemma 5.5. (Robustness of the control law). For every δ > 0, there ex-934
ist T1, T2 <∞ such that when Stages 1 and 2 are terminated at t1 > T1 and t2 > T2935
respectively, we have that limt→∞ ‖ρ(t, ·)− ρ∗‖L2(M(t1)) < δ.936
Proof. In Stage 1, it follows from Theorem 5.3 on the convergence to the desired937
spatial domain that limt→∞M(t) = M∗. Then for every ε1 > 0, we have T1 <∞, such938
25
This manuscript is for review purposes only.
that dH(M(t),M∗) < ε1 for all t > T1, where dH is the Hausdorff distance between939
two sets; see (1). (Note that any appropriate notion of distance can alternatively be940
used here.) Let Stage 1 be terminated at t1 > T1, which implies that the swarm is941
distributed across the domain M(t1). In Stage 2, it follows from Lemma 5.1 on the942
convergence of the heat flow equation to the harmonic map, that for a domain M(t1),943
we have that limt→∞R(t, ·) = ΨM(t1) pointwise, where ΨM(t1) is the harmonic map944
from M(t1) to N (the unit disk). Then, for every ε2 > 0, we have a T2 < ∞, such945
that ‖R(t, ·) − ΨM(t1)‖∞ < ε2 for all t > T2. Let Stage 2 be terminated at t2 > T2,946
which implies that the map from the spatial domain to the disk is R(t2, ·). In Stage 3,947
it follows from the arguments in the proof of Theorem 5.4 (on the convergence to the948
desired density distribution) that limt→∞ ρ(t, ·) = p∗ R(t2, ·) a.e. in M(t1) if the949
map at the end of Stage 2 is R(t2, ·). We characterize the error as limt→∞ ‖ρ −950
ρ∗‖L2(M(t1)) = ‖p∗ R(t2, ·) − p∗ Ψ∗‖L2(M(t1)) = ‖p∗ R(t2, ·) − p∗ ΨM(t1) + p∗ 951
ΨM(t1)−p∗ Ψ∗‖L2(M(t1)) ≤ ‖p∗ R(t2, ·)−p∗ ΨM(t1)‖L2(M(t1)) +‖p∗ ΨM(t1)−p∗ 952
Ψ∗‖L2(M(t1)). Recall that ‖R(t2, ·)−ΨM(t1)‖∞ < ε2, and since p∗ is Lipschitz, we can953
get the bound ‖p∗ R(t2)− p∗ ΨM(t1)‖L2(M(t1)) < δ1 = cε2 (where c is the Lipschitz954
constant times the area of M(t1)). The harmonic map also depends continuously on955
its domain [15], which yields the bound ‖ΨM(t1)−Ψ∗‖∞ < ε3, since dH(M(t1),M∗) <956
ε1. Thus, we get another bound ‖p∗ ΨM(t1) − p∗ Ψ∗‖L2(M(t1)) < δ2 = cε3, and957
that ‖ρ − ρ∗‖L2(M(t1)) < δ1 + δ2 = δ. Therefore, going backwards, for all δ > 0, we958
can find T1 and T2 such that the density error is bounded by δ, when the Stages 1959
and 2 are terminated at t1 > T1 and t2 > T2 respectively.960
5.4. Discrete implementation. In this section, we present consistent schemes961
for discrete implementation of the distributed control laws (32) and (35), where the962
key aspect is the computation of spatial gradients (of ρ in Stage 1, and of ρ, Ψ∗ and963
the components of velocity v in Stage 3). The network graph underlying the swarm is964
a random geometric graph, where the nodes are distributed according to the density965
distribution over the spatial domain. According to this, every agent communicates966
with other agents within a disk of given radius (say r) determined by the hardware967
capabilities, which reduces to the graph having an edge between two nodes if and968
only if the nodes are separated by a distance less than r. We recall the earlier stated969
assumption that the agents know the true x- and y-directions.970
5.4.1. On the computation of p∗. We first begin with an approach to compute971
offline the map p∗ via interpolation. Let the desired domain M∗ ∈ R2 be discretized972
into a uniform grid to obtain M∗d = r1, . . . , rm (the centers of finite elements, where973
rk = (xk, yk)). The desired density ρ∗ : M∗ → R>0 is known, and we compute the974
value of ρ∗ on M∗d to get ρ∗(r1, . . . , rm) = (ρ∗1, . . . , ρ∗m). We also have Ψ∗(x, y) =975
(X∗, Y ∗) ∈ N , for all (x, y) ∈ M∗. Now, computing the integral with respect to the976
Dirac measure for the set M∗d , we obtain Ψ∗(r1, . . . , rm) = (Ψ∗1, . . . ,Ψ∗m). The value of977
the function p∗ at any (X,Y ) ∈ N can be obtained from the relation p∗(Ψ∗1, . . . ,Ψ∗m) =978
ρ∗(r1, . . . , rm) for k = 1, . . . ,m by an appropriate interpolation.979
(ρ∗1, . . . , ρ∗m) = p∗(Ψ∗1, . . . ,Ψ
∗m)
(r1, . . . , rm) (Ψ∗1, . . . ,Ψ∗m)
ρ∗
Ψ∗
p∗
26
This manuscript is for review purposes only.
5.4.2. Discrete control law. As stated earlier, for the discrete implementation980
of the distributed control laws (32) and (35), the key aspect is the computation of981
spatial gradients (of ρ in Stage 1, and of ρ, Ψ∗ and the components of velocity v in982
Stage 3). In the subsequent sections we present two alternative, consistent schemes983
for computing the spatial gradient (of any smooth function, with the above being the984
ones of interest), one using the Jacobian of the harmonic map and the other without985
it.986
Computing the Jacobian of the harmonic map. Let J(r) = ∇Ψ(r) be the987
(non-singular) Jacobian of the harmonic diffeomorphism Ψ : M → N . When the988
steady-state is reached in the pseudo-localization algorithm (30) (i.e., Xi(t + 1) =989
Xi(t) = ψi1 and Yi(t+ 1) = Yi(t) = ψi2), we have, ∀ i ∈ S:990
∑
j∈Ni
1
dj(ψj1 − ψi1) = 0,
∑
j∈Ni
1
dj(ψj2 − ψi2) = 0,991
992
where i is the index of the agent located at r ∈ M and Ni is the set of agents in a993
disk-shaped neighborhood Bε(r) of area ε centered at r. Rewriting the above, we get,994
∀ i ∈ S:995
ψi1 =
∑j∈Ni
1djψj1∑
j∈Ni1dj
, ψi2 =
∑j∈Ni
1djψj2∑
j∈Ni1dj
.(36)996
997
We assume that the agents have the capability in their hardware to perturb the disk of998
communication Bε(r) (by moving an antenna, for instance). The Jacobian J = ∇Ψ999
is computed through perturbations to Ni (i.e., the neighborhood Bε(r)) and using1000
consistent discrete approximations:1001
∂xψ1 ≈ψ1(r + δxe1)− ψ1(r)
δx, ∂yψ1 ≈
ψ1(r + δye2)− ψ1(r)
δy,1002
1003
and similarly for ψ2. Now, ψ1(r + δxe1) is computed as in (36) for N δxi , the set of1004
agents in Bε(r + δxe1) and ψ1(r + δye2) from Bε(r + δye2).1005
Computing the spatial gradient of a smooth function using the Jacobian1006
of Ψ. Let ∇ = (∂x, ∂y) and ∇ = (∂ψ1 , ∂ψ2), where Ψ = (ψ1, ψ2). We have ∂x =1007
(∂xψ1)∂ψ1+ (∂xψ2)∂ψ2
and ∂y = (∂yψ1)∂ψ1+ (∂yψ2)∂ψ2
. Therefore, ∇ = J>∇. For a1008
smooth function f : M → R, we have, ∇f = J>∇f , and the agents can numerically1009
compute ∇ by:1010
(∂f
∂ψ1
)
i
≈ 1
|Ni|∑
j∈Ni
fj − fiψj1 − ψi1
,
(∂f
∂ψ2
)
i
≈ 1
|Ni|∑
j∈Ni
fj − fiψj2 − ψi2
,1011
1012
where i is the index of the agent located at r ∈ M and Ni is the set of agents in a1013
ball Bε(r).1014
Computing the spatial gradient of a smooth function without the Ja-1015
cobian of Ψ. In the absence of a Jacobian estimate, we use the following alternative1016
method for computing an approximate spatial gradient estimate of a smooth function.1017
This is used in Stage 1 of the self-organization process.1018
Let f(r) be the mean value of f over a ball Bε(r):1019
f(r) =1
ε
∫
Bε(r)
fdµ ≈ 1
|Ni|∑
j∈Ni
fj .1020
1021
27
This manuscript is for review purposes only.
We have:1022
1
ε
∂f
∂x≈ 1
ε
f(r + δxe1)− f(x)
δx=
1
ε
∫Bε(r+δxe1)
fdµ−∫Bε(r)
fdµ
δx1023
=1
ε
∫
Bε(r)
(f(r + δxe1)− f(r))
δxdµ1024
≈ 1
ε
∫
Bε(r)
∂f
∂xdµ =
(∂f
∂x
).1025
1026
Similarly,1027
1
ε
∂f
∂y≈ 1
ε
f(r + δye2)− f(x)
δy≈(∂f
∂y
).1028
1029
In all, for any scalar function f , each agent can use the approximation1030
(∇f)i ≈((
∂f
∂x
),
(∂f
∂y
))=
1
ε
(∂f
∂x,∂f
∂y
),(37)1031
1032
to estimate of the gradient ∇f .1033
5.4.3. On the convergence of the discrete system. We have noted earlier1034
that the pseudo-localization algorithm (30) satisfies the consistency condition in that1035
as N → ∞, Equation (30) converges to the PDE (27). The pseudo-localization1036
algorithm is also essentially a weighted Laplacian-based distributed algorithm that is1037
stable. Thus, by the Lax Equivalence theorem [25], the solution of (30) converges to1038
the solution of (27) as N →∞. However, for the distributed control laws in Stages 1-1039
3, we are only able to provide consistent discretization schemes. The dynamics of the1040
swarm (31) with the control laws (32) and (33) are nonlinear for which is no equivalent1041
convergence theorem. Further analysis to determine convergence is required, which1042
falls out the scope of this present work.1043
6. Numerical simulations. In this section, we present numerical simulations1044
of swarm self-organization, that is, of the control laws presented in Sections 4.2 and1045
of Section 5.3.1046
6.1. Self-organization in one dimension. In the simulation of the 1D case,1047
we consider a swarm of N = 10000 agents, the desired density distribution is given by1048
ρ∗(x) = a sin(x) + b, where a = 1− π2N and b = 1
N , x ∈[0, π2
]. We use a kernel-based1049
method to approximate the continuous density function, which is given by:1050
ρ(t, r) =∑
i∈SK
(‖r− ri(t)‖d
),1051
1052
where1053
K(x) =
cddn , for 0 ≤ x < 1,
0, for x ≥ 1,1054
1055
is a flat kernel and cd ∈ R>0 is a constant [8]. We discretize the spatial domain1056
with ∆x = 0.001 units, and use an adaptive time step. The self-organization begins1057
from an arbitrary initial density distribution. Figure 2 shows the initial density dis-1058
tribution, an intermediate distribution and the final distribution. We observe that1059
there is convergence to the desired density distribution, even with noisy density mea-1060
surements.1061
28
This manuscript is for review purposes only.
Algorithm 2 Self-organization algorithm for 2D environments
1: Input: M∗, ρ∗ and k1, k2, K (number of iterations for each stage), ∆t (timestep)
2: Requires:3: Offline computation of p∗ as outlined in Section 5.4.14: Boundary agents are aware of being at boundary or interior of domain, can5: communicate with others along the boundary, can approximate the normal6: to the boundary, and can measure density of boundary agents,7: Agents have knowledge of a common orientation of a reference frame8: Initialize: ri (Agent positions), vi = 0 (Agent velocities)9: Boundary agents localize as outlined in Section 5.1
10: Stage 1:11: for k := 1 to k1 do12: if agent i is at the interior of domain then
13: compute vi(k) = − (∇ρ)iρi
(k) from (32), with (∇ρ)i(k) as in (37),
14: move ri(k + 1) = ri(k) + vi(k)∆t15: else if agent i is at the boundary of domain then16: compute vi(k+1) = vi(k)− (ri(k)−r∗i (k)+vi(k))∆t from (32), and move
ri(k + 1) = ri(k) + vi(k)∆t
17: Stage 2:18: Boundary agents map themselves onto unit circle according to (23)19: for k := 1 to k2 do20: for agent i in the interior do21: compute Xi(k + 1), Yi(k + 1) according to (30)
22: Stage 3:23: for k := 1 to K do24: for agent i in the interior do25: compute vi(k+1) = vi(k)+(−ρi(k)(∇(ρ−p∗Ψ∗))i(k)+(vi(k)·∇)vi(k)−
vi(k))∆t from (33), with (∇(ρ− p∗ Ψ∗))i(k) as in (37)26: update ri(k + 1) = ri(k) + vi(k)∆t
6.2. Self-organization in two dimensions. In the simulation of the 2D case,1062
we first present in Figure 3 the evolution of the boundary of the swarm in Stage 1,1063
where the swarm converges to the target spatial domain M∗ from an initial spatial1064
domain. The target spatial domain, a circle of radius 0.5 units, given by M∗ =1065
(x, y) ∈ R2 | (x − 0.6)2 + y2 ≤ 0.25, with the desired density distribution ρ∗ given1066
by ρ∗(x, y) = 1((x−0.4)2+y2)0.3
. We present in Figures 4 and 5 the result of imple-1067
mentation of the pseudo-localization algorithm with the steady state distributions1068
of Ψ∗ = (ψ∗1 , ψ∗2) respectively. We note that the steady state distribution Ψ∗ as a1069
function of the spatial coordinates (x, y) in this case is linear. Next, we focus on1070
Stage 3 of the self-organization process, where the agents already distributed over the1071
target spatial domain, converge to the desired density distribution. The initial density1072
distribution of the swarm is uniform, and the distributed control law of Stage 3 in1073
Section 5.3, following the discretization scheme outlined in Section 5.4 is implemented.1074
Figure 6 shows the density distribution at a few intermediate time instants of imple-1075
mentation and figure 7 shows the spatial density error plot, where e(ρ) =∫M∗ |ρ−ρ∗|21076
is the spatial density error. The results show convergence as desired.1077
29
This manuscript is for review purposes only.
Fig. 2: Density ρ(x) plotted against position x at different instants of time.
Fig. 3: Evolution of the swarm boundary in Stage 1.
7. Conclusions. In this paper, we considered the problem of self-organization1078
in multi-agent swarms, in one and two dimensions, respectively. The primary contri-1079
bution of this paper is the analysis and design of position and index-free distributed1080
30
This manuscript is for review purposes only.
Fig. 4: Steady-state distribution of ψ∗1 . Fig. 5: Steady-state distribution of ψ∗2 .
Fig. 6: Evolution of density distribution in Stage 3.
control laws for swarm self-organization for a large class of configurations. This was1081
accomplished through the introduction of a distributed pseudo-localization algorithm1082
that the agents implement to find their position identifiers, which then use in their1083
control laws. The validation of the results for more general non-simply connected do-1084
mains will be considered in the future. An extension to this work will involve the char-1085
31
This manuscript is for review purposes only.
Fig. 7: Spatial density error e(ρ) =∫M∗ |ρ− ρ∗|2 vs time,
acterization of constraints on the local density function to capture finite robot sizes1086
and collision avoidance constraints, as well as accounting for possible non-holonomic1087
constraints on the motion of the robots.1088
Acknowledgments. The authors would like to thank Prof. Lei Ni at the UC1089
San Diego Mathematics Department and the reviewers of this manuscript for their1090
valuable inputs.1091
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[6] S. Camazine, Self-organization in biological systems, Princeton University Press, 2003.1104[7] I. Chattopadhyay and A. Ray, Supervised self-organization of homogeneous swarms using1105
ergodic projections of markov chains, IEEE Transactions on Systems, Man, & Cybernetics.1106Part B: Cybernetics, 39 (2009).1107
[8] Y. Cheng, Mean shift, mode seeking, and clustering, IEEE Transactions on Pattern Analysis1108and Machine Intelligence, 17 (1995).1109
[9] A. J. Chorin and J. E. Marsden, A mathematical introduction to fluid mechanics, vol. 3,1110Springer, 1990.1111
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